Let G/H be a reductive homogeneous space. In all known examples, if
G/H admits compact Clifford-Klein forms, then it admits "standard"
ones, by uniform lattices of some reductive subgroup L of G acting
properly on G/H. In order to obtain more generic Clifford-Klein forms,
we prove that for L of real rank 1, if one slightly deforms in G a
uniform lattice of L, then its action on G/H remains properly
discontinuous. As an application, we obtain compact quotients of SO(2,2n)/U(1,n) by Zariski-dense discrete subgroups of SO(2,2n) acting properly discontinuously.

(GCOE Lectures)

Date:

February 17 (Wed), 2010, 10:30-11:30

February 17 (Wed), 2010, 15:00-16:00

February 18 (Thu), 2010, 10:30-11:30

February 18 (Thu), 2010, 15:00-16:00

February 19 (Fri), 2010, 10:30-12:00

Place:

Room 126, Graduate School of Mathematical Sciences, the University of Tokyo

In this course I will focus on recent advances
on our understanding of discrete subgroups of Lie groups.

I will first survey how ideas from semisimple algebraic groups,
ergodic theory and representation theory help us to understand properties of these discrete subgroups.

I will then focus on a joint work with Jean-Francois Quint
studying the dynamics of these discrete subgroups on finite volume homogeneous spaces and proving the following result:

We fix two integral matrices A and B of size d, of determinant 1,
and such that no finite union of vector subspaces is invariant by A and B.
We fix also an irrational point on the d-dimensional torus. We will then prove that for n large the set of images of this point by the words in A and B of length at most n becomes equidistributed in the torus.

Date:

April 6 (Tue), 2010, 16:30-18:00

Place:

Room 126, Graduate School of Mathematical Sciences, the University of Tokyo

Speaker:

Shu Kato (加藤 周) (Kyoto University)

Title:

On the characters of tempered modules of affine Hecke algebras of classical type

We present an inductive algorithm to compute the characters
of tempered modules of an affine Hecke algebras of classical
types, based on a new class of representations which we call
"tempered delimits". They have some geometric origin in the
eDL correspondence.

Our new algorithm has some advantage to the Lusztig-Shoji
algorithm (which also describes the characters of tempered
modules via generalized Green functions) in the sense it
enables us to tell how the characters of tempered modules
changes as the parameters vary.

This is a joint work with Dan Ciubotaru at Utah.

Date:

April 15 (Thu), 2010, 16:30-18:00

Place:

Room 056, Graduate School of Mathematical Sciences, the University of Tokyo

A new type of hypergeometric differential equations was introduced and studied by H. Sekiguchi. The investigated system of partial differential equation generalizes the Gauss-Aomoto-Gelfand system which in its turn stems from the classical set of differential relations for the solutions to a generic algebraic equation introduced by K.Mayr in 1937. Gauss-Aomoto-Gelfand systems can be expressed as the determinants of $2\times 2$ matrices of derivations with respect to certain variables. H. Sekiguchi generalized this construction by looking at determinations of arbitrary $l\times l$ matrices of derivations with respect to certain variables.

In this talk we study the dimension of global (and local) solutions to the generalized systems of Gauss-Aomoto-Gelfand hypergeometric systems. The main results in the talk are a combinatorial formula for the dimension of global (and local) solutions of the generalized Gauss-Aomoto-Gelfand system and a theorem on generic holonomicity of a certain class of such systems.

Date:

April 20 (Tue), 2010, 16:30-18:00

Place:

Room 126, Graduate School of Mathematical Sciences, the University of Tokyo

We study the restriction of Vogan-Zuckerman derived functor
modules Aq(λ) to symmetric subgroups. An algebraic condition
for the discrete decomposability of Aq(λ) was given by
Kobayashi, which offers a framework for the detailed study
of branching law. In this talk, when Aq(λ) is discretely decomposable,
we construct some of irreducible components occurring
in the branching law and determine their associated
variety.

Date:

May 11 (Tue), 2010, 16:30-18:00

Place:

Room 126, Graduate School of Mathematical Sciences, the University of Tokyo

Speaker:

Hisayosi Matumoto (松本久義) (the University of Tokyo)

Title:

On a finite $W$-algebra module structure on the space of continuous Whittaker vectors for an irreducible Harish-Chandra module

Let $G$ be a real reductive Lie group. The space of continuous Whittaker vectors for an irreducible Harish-Chandra module has a structure of a module over a finite $W$-algebra. We have seen such modules are irreducible for groups of type A. However, there is a counterexample to the naive conjecture. We discuss a refined version of the conjecture and further examples in this talk.

Date:

May 18 (Tue), 2010, 16:30-18:00

Place:

Room 126, Graduate School of Mathematical Sciences, the University of Tokyo

Speaker:

Birgit Speh (Cornel University)

Title:

On the eigenvalues of the Laplacian on locally symmetric hyperbolic spaces

A famous Theorem of Selberg says that the non-zero eigenvalues of the Laplacian acting on functions on quotients of the upper half plane H by congruence subgroups of the integral modular group, are bounded away from zero, as the congruence subgroup varies. Analogous questions on Laplacians acting on differential forms of higher degree on locally symmetric spaces (functions may be thought of as differential forms of degree zero) have geometric implications for the cohomology of the locally symmetric space.

Let $X$ be the real hyperbolic n-space and $\Gamma \subset $ SO(n, 1) a congruence arithmetic subgroup. Bergeron conjectured that the eigenvalues of the Laplacian acting on the differential forms on $ X / \Gamma $ are bounded from the below by a constant independent of the congruence subgroup. In the lecture I will show how one can use representation theory to show that this conjecture is true provided that it is true in the middle degree.

This is joint work with T.N. Venkataramana.

Date:

May 25 (Tue), 2010, 17:00-18:00

Place:

Room 126, Graduate School of Mathematical Sciences, the University of Tokyo

The theory of endoscopy came out of the Langlands functoriality and the trace formula.
In this talk, I will briefly explain what the endoscopy is, and talk about packet, formal degree and Whittaker normalization of transfer.
I would like to talk about the connection between these topics and the endoscopy.

(GCOE Lectures)

Date:

June 1 (Tue), 2010, 16:30-18:00

Place:

Room 126, Graduate School of Mathematical Sciences, the University of Tokyo

I will give an introduction to the cohomology of noncompact locally symmetric spaces $X_\Gamma =K \backslash G / \Gamma $.
If $X_\Gamma $ is cocompact this cohomology can be expressed as the $(g,K)$-cohomology of automorphic representations. I will explain how representation theory and automorphic forms can be used to study the cohomology in this case.

(GCOE Lectures)

Date:

June 3 (Thu), 2010, 16:30-18:00

Place:

Room 470, Graduate School of Mathematical Sciences, the University of Tokyo

I will give an introduction to the cohomology of noncompact locally symmetric spaces $X_\Gamma =K \backslash G / \Gamma $.
If $X_\Gamma $ is cocompact this cohomology can be expressed as the $(g,K)$-cohomology of automorphic representations. I will explain how representation theory and automorphic forms can be used to study the cohomology in this case.

Let G^ be a simple Lie group of Hermitian type and U^ be a maximal parabolic subgroup of G^ with abelian nilradical. The flag manifold M^= G^/ U^ is the Shilov
boundary of an irreducible bounded symmetric domain of tube type. M^ has the G-invariant causal structure. A causal Makarevich space is, by definition, an open symmetric G-orbit M in M^, endowed with the causal structure induced from that
of the ambient space M^, G being a reductive subgroup of G^. All symmetric cones fall in the class of causal Makarevich spaces.
In this talk, we determine the causal automorphism groups of all causal Makarevich spaces.

Date:

July 15 (Thu), 2010, 14:30-16:00

Place:

Room 122, Graduate School of Mathematical Sciences, the University of Tokyo

The irreducible rational representations of the complex orthogonal
group $\mathrm{O}_n$ are labeled by a certain set of Young diagrams,
and we denote the representation corresponding to the Young diagram
$D$ by $\sigma^D_n$. Consider the tensor product
$\sigma^D_n\otimes\sigma^E_n$ of two such representations. It can
be decomposed as
\[\sigma^D_n\otimes\sigma^E_n=\bigoplus_Fm_F\sigma^F_n,\]
where for each Young diagram $F$ in the sum, $m_F$ is the
multiplicity of $\sigma^F_n$ in $\sigma^D_n\otimes\sigma^E_n$. In
the case when the Young diagram $E$ consists of only one row, a
description of the multiplicities in $\sigma^D_n\otimes\sigma^E_n$
is called the {\em Pieri Rule}. In this talk, I shall describe a
family of algebras whose structure encodes a generalization of the
Pieri Rule.

Date:

September 1 (Wed), 2010, 16:30-18:00

Place:

Room 002, Graduate School of Mathematical Sciences, the University of Tokyo

Groups of Kac-Moody type are natural generalizations of Kac-Moody groups over fields in the sense that they have an RGD-system. This is a system of subgroups indexed by the roots of a root system and satisfying certain commutation relations.
Roughly speaking, there is a one-to-one correspondence between groups of Kac-Moody type and Moufang twin buildings. This correspondence was used in the last decade to prove several group theoretic results on RGD-systems and in particular on Kac-Moody groups over fields.

In my talk I will explain RGD-systems and how they provide twin
buildings in a natural way. I will then present some of the group theoretic applications mentioned above and describe how twin buildings are used as a main tool in their proof.

Room 126, Graduate School of Mathematical Sciences, the University of Tokyo

Speaker:

Bernhard Mühlherr (Justus-Liebig-Universität Giessen)

Title:

Mini-course on buildings

Abstract:

The goal of this course is to provide an overview on the theory of
buildings which was developed by Jacques Tits. The idea is to start
with basic examples, then to explain the classification of spherical
buildings and to finish with a survey on affine buildings and twin
buildings.

1st Lecture: We recall the basic facts about Coxeter groups which are
important for the theory of buildings and we explain the class of rank 2
buildings which are generalized polygons or trees. Finally we give a
general definition of a building and some characterizations.

2nd Lecture: We start with the basic facts about spherical buildings and
the Moufang property. We explain Tits' extension theorem for spherical
buildings an how it implies the Moufang property. We then give an
outline of the classification.

3rd Lecture: We give examples of affine buildings and a survey on the
main ideas in their theory and their classification. We then move on to
the theory of twin buildings and explain how they can be used to study
groups of Kac-Moody type. We will finish with some remarks on the
classification problem for arbitrary buildings.

Date:

October 26 (Tue), 2010, 16:30-18:00

Place:

Room 126, Graduate School of Mathematical Sciences, the University of Tokyo

Speaker:

Daniel Sternheimer (Keio University and Institut de Mathématiques de Bourgogne)

Title:

Some instances of the reasonable effectiveness (and limitations) of symmetries and deformations in fundamental physics

In this talk we survey some applications of group theory and deformation theory (including quantization) in mathematical physics. We start with sketching applications of rotation and discrete groups representations in molecular physics ("dynamical" symmetry breaking in crystals, Racah-Flato-Kibler; chains of groups and symmetry breaking). These methods led to the use of "classification Lie groups" ("internal symmetries") in particle physics. Their relation with space-time symmetries will be discussed. Symmetries are naturally deformed, which eventually brought to Flato's deformation philosophy and the realization that quantization can be viewed as a deformation, including the many avatars of deformation quantization (such as quantum groups and quantized spaces). Nonlinear representations of Lie groups can be viewed as deformations (of their linear part), with applications to covariant nonlinear evolution equations. Combining all these suggests an Ansatz based on Anti de Sitter space-time and group, a deformation of the Poincare group of Minkowski space-time, which could eventually be quantized, with possible implications in particle physics and cosmology. Prospects for future developments between mathematics and physics will be indicated.

Date:

November 2 (Tue), 2010, 16:30-18:00

Place:

Room 126, Graduate School of Mathematical Sciences, the University of Tokyo

Harmonic functions are real-analytic and so automatically extend from being functions of real variables to being functions of complex variables. But how far do they extend? This question may be answered by twistor theory, the Penrose transform, and associated geometry. I shall base the constructions on a formula of Bateman from 1904. This is joint work with Feng Xu.

(GCOE Lectures)

Date:

November 5 (Fri), 2010, 16:30-18:00

Place:

Room 123, Graduate School of Mathematical Sciences, the University of Tokyo

The geodesics on a Riemannian manifold form a distinguished family of curves, one in every direction through every point. Sometimes two metrics can provide the same family of curves: the Euclidean metric and the round metric on the hemisphere have this property. It is also possible that a family of curves does not arise from a metric at all. Following a classical procedure due to Roger Liouville, I shall explain how to tell these cases apart on a surface. This is joint work with Robert Bryant and Maciej Dunajski.

(GCOE Lectures)

Date:

November 8 (Mon), 2010, 16:30-18:00

Place:

Room 128, Graduate School of Mathematical Sciences, the University of Tokyo

The circle is acted upon by the rotation group SO(2) and there are plenty of differential operators invariant under this action. But the circle is also acted upon by SL(2,R) and this larger symmetry group cuts down the list of invariant differential operators to something smaller but more interesting! I shall explain what happens and how this phenomenon generalises to spheres. These constructions are part of a general theory but have numerous unexpected applications, for example in suggesting a new stable finite-element scheme in linearised elasticity (due to Arnold, Falk, and Winther).

Date:

December 21 (Tue), 2010, 16:30-18:00

Place:

Room 126, Graduate School of Mathematical Sciences, the University of Tokyo

Speaker:

Katsuyuki Naoi (直井克之) (The University of Tokyo)

Title:

Some relation between the Weyl module and the crystal basis of the tensor product of fudamental representations

The Lie algebra defined by the tensor product of a simple Lie algebra and a polynomial ring is called the current algebra, and the Weyl module is defined by a finite dimensional module of the current algebra having some universal property.
The fundamental representation is a irreducible, finite dimensional, level zero integrable representation of the quantized affine algebra, and it is known that this module has a crystal basis.
If the simple Lie algebra is of ADE type, Fourier and Littelamnn has shown that the Weyl module is isomorphic to a module called the Demazure module.
Using this fact, we can easily see that the (Z-graded) characters of the Weyl module and the crystal basis of the tensor product of fundamental representations coincides.
In my talk, I will introduce the generalization of this result in the non-simply laced case.
In this case, the result of Fourier and Littelmann does not necessarily true, but we can show the characters of two objects also coincide in this case.
This fact is shown using the Demazure modules and its ''crystal basis'' called the Demazure crystals.